Wednesday, June 11, 2014

$\kappa$-presentable structures

Our topic in this post is Definition 5.1 in the Abraham-Magidor Handbook Article [AM]

Assume $\chi$ is some sufficiently large regular cardinal, and $\langle H(\chi),\in, <_\chi\rangle$ is as usual.

An elementary substructure $M$ of $H(\chi)$ is $\kappa$-presentable for a regular cardinal $\kappa$ if $M=\bigcup_{i<\kappa} M_i$ for some sequence $\langle M_i:i<\lambda\rangle$ where
  1. Each $M_i$ is an elementary substructure of $H(\chi)$,
  2. $i<j<\lambda\Longrightarrow M_i\subseteq M_j$, 
  3. if $\delta<\lambda$ is a limit, then $M_\delta=\bigcup_{i<\delta} M_i$,
  4. $M$ has cardinality $\kappa$,
  5. $\kappa+1\subseteq M$, and
  6. $M_i\in M_{i+1}$ for each $i<\kappa$ (so $M_i\in M_j$ whenever $i<j<\kappa$)
No assumption is made on the cardinality of $M_i$ for $i<\kappa$, so each $M_i$ could have cardinality $\kappa$, or cardinality less than $\kappa$.

This is clearly related to the notion of a structure being internally approachable of length $\kappa$; this concept suffices for the [AM] presentation, but I'm not certain it suffices for what Shelah is doing in [Sh:371].  This is one of the details that I'd like to clear up.

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